Samuel has been experimenting with quadratic equations of the form: y=ax²+2bx+c Samuel chose values of a, b and c by taking three consecutive terms from the sequence: 1,2,4,8,16,32... Example quadratic equations might be:
y=x²+4x+4 or y=2x²+8x+8
Try plotting some graphs based on Samuel's quadratic equations, for different sets of consecutive terms from his sequence.
Do you notice anything interesting?
Can you make any generalisations? Can you prove them?
Samuel's sequence is an example of a geometrical sequence, created by taking a number and then repeatedly multiplying by a common ratio. Samuel's sequence starts at1 and has common ratio2 (each number in the sequence is 2 times the previous number).
Create some more geometrical sequences and substitute consecutive terms into Samuel's quadratic equation.
Here are some questions you might like to explore:
Can you make any predictions about the graph from the geometric sequence you use to generate the equation?
What if the common ratio is a fraction, or a negative number?
What if the starting number for your geometric sequence is a fraction, or a negative number?
Can you make any generalisations? Can you prove them?
Parabolas created using Geometric Sequences
Samuel has been experimenting with quadratic equations of the form:
y=ax² +2bx+c
Samuel chose values of a, b and c by taking three consecutive terms from the sequence:
1,2,4,8,16,32...
Example quadratic equations might be:
y=x²+4x+4 or y=2x²+8x+8
Try plotting some graphs based on Samuel's quadratic equations, for different sets of consecutive terms from his sequence.
Do you notice anything interesting?
Can you make any generalisations? Can you prove them?
Samuel's sequence is an example of a geometrical sequence, created by taking a number and then repeatedly multiplying by a common ratio. Samuel's sequence starts at 1 and has common ratio 2 (each number in the sequence is 2 times the previous number).
Create some more geometrical sequences and substitute consecutive terms into Samuel's quadratic equation.
Here are some questions you might like to explore:
Can you make any predictions about the graph from the geometric sequence you use to generate the equation?
What if the common ratio is a fraction, or a negative number?
What if the starting number for your geometric sequence is a fraction, or a negative number?
Can you make any generalisations? Can you prove them?